Method for determining the slip angles and/or the cornering forces of a braked vehicle

ABSTRACT

A method is described for determining the slip angles α i  and/or the cornering forces F S  of a braked vehicle. Starting from a simplified vehicle model and using the wheel speeds V Ri , the steering angle β, the yaw rate φ and the master brake cylinder pressure P HB2  or the wheel brake pressure P i  as measured variables, the desired variables are determined as estimated variables.

PRIOR ART

The slip angles of the wheels or of the axles of a braked motor vehicle are important variables for the vehicle dynamics control of a motor vehicle. This also applies to the cornering forces on the individual wheels. The two variables can be measured directly or determined only with difficulty.

SUMMARY OF THE INVENTION

The invention proposes an estimate of the above mentioned two variables, i.e., the slip angles and the cornering forces there being included in the estimate measured variables which are required in any case for vehicle dynamics control, and are therefore measured.

The present invention generally provides a method for increasing the controllability of a wheeled vehicle having a dynamic drive control system for controlling at least the brake pressure of the wheels, and including the steps of determining slip angle values α_(i), where i=1, . . . 4 and indicates the respective vehicle wheel, and applying the determined slip angle values α_(i) to the dynamic drive control system to improve the vehicle control; and wherein the step of determining the slip angle values α_(i) comprises: determining respective wheel speeds V_(Ri), the steering angle δ, the vehicle yaw rate ρ and respective wheel brake cylinder pressures P_(Ri) ; calculating braking forces F_(Bi) from the wheel brake pressures P_(Ri) according to the equation ##EQU1## where θR is the amount of inertia of the wheel, C_(Pi) is the braking ratio, and R is the radius of the wheel; determining the longitudinal vehicle speed V_(L) from the wheel speeds V_(Ri) and from the deceleration of the vehicle; estimating the vehicle transverse speed V_(q) from the determined values for δ_(i) and ρ using a Kalman Filter; estimating the attitude angles β_(v) and β_(H) from the estimated value V_(q) of the vehicle transverse speed V_(q), according to the relationships ##EQU2## where V and H indicate the front and rear, respectively, and where 1₁ and 1₂ are the respective lengths of the vehicle axles from the vehicle center of gravity; and determining the slip angles α_(i), using the estimated attitude angle values β_(v) and β_(H) and values δE_(i) corresponding the respective elastic steering angle values, according to the relationships

    α.sub.1 =δ-β.sub.v -δ.sub.E1

    α.sub.2 =δ-β.sub.v -δ.sub.E2

    α.sub.3 =β.sub.H -δ.sub.E3

    α.sub.4 =β.sub.H -δ.sub.E4

According to the preferred embodiment of the invention at least the wheel speeds V_(Ri), the steering angle δ and the yaw rate ρ are measured or determined from measured values of the master brake cylinder pressure. The wheel brake cylinder pressures P_(Ri) maybe measured. Moreover, preferably the cornering forces F_(si) on the respective wheels are determined using the estimated attitude angle values β_(v) and β_(H) and the steering angle δ and the determined cornering forces F_(si) are applied to the dynamic drive control system to further improve the controllability of the vehicle.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1a and 1b are simplified vehicle models used to explain the present invention.

FIGS. 2a, 2b and 3 are further diagrams used in explaining the present invention.

FIGS. 4-7 show a comparison between measured and estimated slip angles of a specific vehicle for various driving maneuvers and different surfaces.

FIG. 8 shows the relationship between the slip angle and the lateral force of a vehicle.

FIG. 9 shows a group of curves illustrating the relationship between the actual and simulated vehicle speed and between the actual and simulated yaw angular velocity when the slip angle is in the non-linear region of the lateral force/slip curve of FIG. 8.

FIG. 10 is a curve illustrating how the difference between the measured and simulated yaw rates can be used, according to a modification of the method according to the invention, to enable to determination of the tire normal forces.

FIG. 11 is a block diagram showing the structure of a slip stiffness controller.

FIG. 12 contains curves showing a comparison between measured and simulated vehicle longitudinal speed using a slip stiffness controller according to FIG. 11.

FIG. 13 is a block diagram of an arrangement for carrying out the method according to the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The point of departure for the further considerations is the simple vehicle model sketched in FIGS. 1a and 1b, in which the pitching, rolling and up-and-down motion are not taken into account. Using these simplifications, the theorems of momentum and angular momentum yield the following equations:

Momentum theorem

    m(V.sub.L +V.sub.q φ)=F.sub.SV sin δ+F.sub.BV cos δ+F.sub.BH)                                         (1.1)

    m(V.sub.q -V.sub.L φ)=F.sub.SV cos δ-F.sub.BV sin δ+F.sub.SH(1.2)

    N.sub.1 +N.sub.2 +N.sub.3 +N.sub.4 -mg=0                   (1.3)

Angular momentum theorem ##EQU3## where

F_(SV) =F_(S1) +F_(S2)

F_(SH) =F_(S3) +F_(S4)

F_(BV) =F_(B1) +F_(B2)

F_(BH) =F_(B3) +F_(B4)

here

m is the vehicle mass

θ_(H) is the moment of inertia about the vertical axis

g is the acceleration due to gravity

V_(L) is the longitudinal vehicle speed

V_(q) is the transverse vehicle speed at the centre of gravity

φ is the yaw rate about the vertical axis

F_(Si) is the cornering force of the i-th wheel

F_(Bi) is the braking force of the i-th wheel

δis the steering angle

N_(i) is the normal force of the i-th wheel

l₁, l₂ are axle distances from the centre of gravity

d is the wheel distance from the longitudinal axis,

and

h is the height of the centre of gravity

Apart from equations (1.3), 1.5), and (1.6), the definite determination of the normal forces N₁, . . . , N₄ requires a further relationship, which is obtained by taking account of the vehicle compression. It holds for small compressions X₁, . . . X₄ that:

    X.sub.1 +X.sub.4 =X.sub.2 +X.sub.3                         (1.7)

Assuming that the spring constants of the wheel suspensions are equal, it follows that:

    N.sub.1 +N.sub.4 =N.sub.2 +N.sub.3                         (1.8)

In the stationary state, it holds for the moments above the steering axis, which for the sake of simplicity is assumed to be perpendicular to the plane of the drawing, that:

    F.sub.S (n.sub.R +n.sub.K)-F.sub.B r.sub.S -C.sub.AS δ.sub.E =0(2.1)

The cornering force and braking force are calculated in accordance with the HSRI tire model: ##EQU4## Here, r_(S) is the steering offset

n_(K) is the kinematic trailing

n_(R) is the tire trailing

δ is the steering angle

δ_(E) is the elastic steering angle

β is the attitude angle

α is the slip angle

V_(x) is the speed of the centre of gravity of the wheel

C_(AS) is the resultant axle and steering stiffness

and ##EQU5## follows from equations (2.2) and (2.3) that: ##EQU6## C.sub.α is the slip stiffness of the tire C.sub.λ is the tire longitudinal stiffness

λ is the tire slip

μ is the coefficient of friction, and

N is the normal force

It follows from equation (2.4), using the approximation tan α≈α, that ##EQU7## or, with

    α=δ-β-δ.sub.E in accordance with FIG. 2b(2.6) ##EQU8## It follows from equation (2.1) and (2.7) that: ##EQU9## where 1.sup.E =n.sub.R +n.sub.K.

Starting from the fundamental equations (2.6), (2.8) and (2.9), the elastic steering angle α_(E) the slip angle α and the cornering forces F_(Si) of the four wheels are yielded as: ##EQU10##

The following abbreviations are used herein: ##EQU11##

Furthermore, we may put: ##EQU12## Here, l_(EV) is (n_(R) +n_(K))_(front) axle

l_(KH) is (n_(R) +n_(K))_(rear) axle

V_(Ri) are the wheel speeds

C_(AS) is the resultant axle and steering stiffness, front axle

r_(SV) is the steering offset at the front axle

C_(A) is the resultant axle stiffness, rear axle, and

r_(SH) is the steering offset at the rear axle

In these relationships, the longitudinal vehicle speed V_(L), the transverse vehicle speed V_(q) and the braking forces F_(Bi) are variables still to be determined, while the steering angle δ, the yaw rate φ and the wheel speeds V_(Ri) are measured variables. The rest are constants.

Applying the angular momentum theorem in accordance with FIG. 3 yields the relationship: ##EQU13## where M_(HR) =C_(P) P_(R), and

θ_(R) is the moment of inertia of the wheel

C_(P) is the braking torque ratio

P_(R) is the wheel brake cylinder pressure

M_(HR) is the braking torque, and

R is the radius of the wheel

The braking force of the i-th wheel is thus yielded as: ##EQU14## The wheel brake cylinder pressures can be estimated in a known way from the measured inlet pressure (master brake cylinder pressure) in conjunction with a hydraulic model, or can be directly measured.

The longitudinal vehicle speed is obtained from the wheel signals and the estimated vehicle retardation: ##EQU15## Substituting equations (2.18)-(2.21) in equations (1.2) and (1.4) yields the relationships: ##EQU16## where ##EQU17##

Finally, substituting equation (2.26) in equations (4.1) and (4.2) yields the system of differential equations: ##EQU18##

Equations (4.3) and (4.4) represent a simple, linear time-variant vehicle model.

Introducing the state variables X₁ =V_(q) and X₂ =φ, and replacing X_(iK) by the approximation ##EQU19## where i=1.2 and

T is the scanning time, yields the discrete state space model:

    X.sub.K+1 =Ψ.sub.K X.sub.K +U.sub.K                    (4.5)

where ##EQU20## Since φ is measured, the model (4.5) can be reduced to the order one. It holds that: ##EQU21## Taking account of the state noise V_(K) and the measurement noise W_(K), (4.6) yields the stochastically disturbed system: ##EQU22## from which an estimated value X_(1K) =V_(qK) is obtained by means of a Kalman filter. It then holds for the estimated attitude angle that ##EQU23##

The slip angles and the cornering forces of the four wheels can thus be determined in accordance with equations (2.14)-(2.21).

The mean slip angles at the front axle and at the rear axle, which are required for the vehicle controller, are yielded as: ##EQU24##

It is decisive for the quality of the estimate of the transverse vehicle speed V_(q) from equation (4.7) that there should already be a good estimated value X₁₀ =V_(q0) before braking. Since in the unbraked case the slip stiffness cannot be calculated from equations (2.22)-(2.25), they will initially be taken as constant.

Assuming that the slip stiffnesses of the wheels of one axle are equal, and designating the resultant slip stiffness of the front axle by C_(VO) and the resultant slip stiffness of the rear axle by C_(HO), according to equations (4.3) and (4.4) the differential equation for the unbraked vehicle is ##EQU25##

It then holds for the lateral forces on the front and rear axles that: ##EQU26##

The gradient C_(RN) is determined by means of a PD control law:

    C.sub.RN =C.sub.RL -K.sub.p |φ.sub.S -φ.sub.n |-K.sub.D |φ.sub.S -φ.sub.n |

where

φ_(n) is the measured yaw angular velocity ##EQU27## is the simulated yaw angular velocity, and are gains >0

The resulting slip stiffnesses C_(V), C_(H) are determined from C_(RV), C_(RH) by means of a simple wheel suspension model.

FIG. 11 shows the structure of the "slip stiffness controller".

FIG. 12 shows a comparison between measured and simulated longitudinal vehicle speed using the "slip stiffness-controller".

Equations (4.10), however, describe the vehicle behaviour satisfactorily only if the slip angles vary in the linear part of the lateral force/slip curve, that is to say if it holds that α≦α_(o) (see FIG. 8).

If the slip angle is in the non-linear region of the lateral force/slip curve (α>α_(o)), serious differences arise between the actual and simulated vehicle speed, and between the actual and simulated yaw angular velocity (see FIG. 9).

Since the yaw rate is measured, the difference between the measured and simulated yaw rates can be used as an indicator for the transition from the linear to the non-linear region of the lateral force/slip curve. As soon as it is detected that the linear region of the lateral force/slip curve has been left, the connection between the lateral force and slip angle is described approximately by a straight line of gradient C_(RN) (see FIG. 10).

It is now also possible to determine in addition the tire normal forces.

The tire normal forces can be obtained from equations (1.3), (1.5), (1.6) and (1.8). It holds that: ##EQU28##

FIGS. 4 to 7 show comparisons between measured (α_(v)) and estimated (α_(v)) slip angles in the case of various driving manoeuvres on different surfaces for a specific vehicle. In detail,

FIG. 4 shows driving on ice+curve

FIG. 5 shows driving on ice+evading manoeuvre

FIG. 6 shows driving in the case of μ split, and

FIG. 7 shows driving on asphalt+curve

An exemplary embodiment of the invention is to be explained with the aid of FIG. 13. Sensors 1 for the wheel speeds V_(Ri), 2 for the wheel cylinder pressure P_(i) or the master brake cylinder pressure P_(HB2), 3 for the steering angle δ, and 4 for the yaw rate φ are provided.

With the aid of the measured variables V_(Ri) and P_(i) or P_(HB2), the braking forces F_(Bi) are determined in a block 5 in accordance with the relationship 3.3, and the longitudinal vehicle speed V_(L) is obtained, for example, by means of the relationship 3.4. The terminals 5a are intended to indicate the input of the constants required to determine the variables.

The estimated variable V_(q) (relationship 4.7) is determined in a further block 6 by means of the measured variables δ and φ and a Kalman filter contained there. The terminals 6a are intended to indicate that the controller of FIG. 11 feeds the variables C_(V) and C_(H) via these.

The output variables of blocks 5 and 6, and the wheel speeds V_(Ri) are fed to a block 7, which determines the attitude angles β_(V) and β_(H) in accordance with relationships 2.26, the wheel slips λ_(i) in accordance with 2.27 and 2.28, the elastic steering angles δ_(Ei) in accordance with the relationships 2.10 to 2.13 and, finally, determines the slip angle α_(i) and/or the cornering forces F_(Si), which it outputs at terminals 7a. Reference is also made here to FIG. 6 plus the associated description of German Patent Application P 40 30 704.2 corresponding to U.S. patent application Ser. No. 07/859,438. 

We claim:
 1. A method for increasing the controllability of a wheeled vehicle having a dynamic drive control system for controlling at least the brake pressure applied to the wheels, said method including the steps of:determining the respective the wheel speeds V_(Ri), where i=1, . . . 4 and indicates the respective vehicle wheel, the steering angle δ, the yaw rate ρ and the respective wheel brake cylinder pressures P_(Ri) ; calculating the braking forces F_(Bi) from the wheel brake pressures P_(Ri) according the equation ##EQU29## where θR is the amount of inertia of the wheel, C_(Pi) is the braking ratio, and R is the radius of the wheel;determining the longitudinal vehicle speed V_(L) from the wheel speeds V_(Ri) and from the deceleration of the vehicle; estimating the vehicle transverse speed V_(q) from the determined values for δ and ρ using a Kalman Filter; estimating the attitude angles β_(v) and β_(v) from the estimated value V_(q) of said vehicle transverse speed V_(q), according to the relationships ##EQU30## where V and H indicate the front and rear, respectively, and where l₁ and l₂ are the respective lengths of the vehicle axles from the vehicle center of gravity; anddetermining the respective elastic steering angle values δE_(i) from the calculated braking forces F_(Bi) ; determining the slip angles α_(i), using said estimated attitude angle values β_(V) and β_(H) and said respective elastic steering angle values δE_(i), according to the relationships

    α.sub. =δ-β.sub. V -δ.sub.E1

    α.sub.2 =δ-β.sub.V -δ.sub.E2

    α.sub.3 =β.sub.H -δ.sub.E3

    α.sub.4 =β.sub.H -δ.sub.E4

and, applying the determined slip angles α_(i) to the dynamic drive control system and utilizing the values α_(i) for controlling the brake pressure applied to the wheels.
 2. Method according to claim 1, further comprising, additionally determining the tire normal forces N_(i).
 3. The method according to claim 1 wherein at least the wheel speeds V_(Ri), the steering angle δ and the yaw rate ρ are measured.
 4. The method according to claim 3 wherein the wheel brake cylinder pressures P_(Ri) are measured.
 5. The method according to claim 3 wherein the master brake cylinder pressure is measured and the respective wheel brake pressures P_(Ri) are determined from the measured master brake cylinder pressure.
 6. The method according to claim 1 further comprising determining the cornering force F_(si) on the respective wheels using the estimated attitude angle values β_(V) and β_(H) and the steering angle δ; and applying the cornering forces F_(si) to the dynamic drive control system to further improve the controllability of the vehicle. 